Extinction times for a general birth, death and catastrophe process

The birth, death and catastrophe process is an extension of the birth-death process that incorporates the possibility of reductions in population of arbitrary size. We will consider a general form of this model in which the transition rates are allowed to depend on the current population size in an arbitrary manner. The linear case, where the transition rates are proportional to current population size, has been studied extensively. In particular, extinction probabilities, the expected time to extinction, and the distribution of the population size conditional on nonextinction (the quasi-stationary distribution) have all been evaluated explicitly. However, whilst these characteristics are of interest in the modelling and management of populations, processes with linear rate coefficients represent only a very limited class of models. We address this limitation by allowing for a wider range of catastrophic events. Despite this generalisation, explicit expressions can still be found for the expected extinction times.

Factors supporting the non-persistence of fruit fly populations in South Australia

For purposes of interstate and international fruit trade, it is necessary to demonstrate that in areas in which fruit fly species have not previously established permanent populations, but which are subject to introductions of fruit flies from outside the area, the introduced population once detected, has not become established. In this paper, we apply methodology suggested mainly by Carey (1991, 1995) to introductions of Mediterranean fruit fly (Medfly), Ceratitis capitata Weid., and Queensland fruit fly (QFF) Bactrocera tryoni Froggatt (Diptera: Tephritidae) to South Australia, a state in which these species do not occur naturally and in which introductions, once detected, are actively treated. By analysing historical data associated with fruit fly outbreaks in South Australia, we demonstrate that: (i) fruit flies occur seasonally, as would occur in established populations, except there is no evidence of the critical spring generation of either species; (ii) there is no evidence of increasing frequency of outbreaks, trapped flies or larval occurrences over 29 years; (iii) there is no evidence of decreasing time between catches of adult flies as the years progress; (iv) there is no decrease in the mean number of years between outbreaks in the same locations; (v) there is no statistically significant recurrence of outbreaks in the same locations in successive years; (vi) there is no evidence of spread of outbreaks outwards from a central location; (vii) the likelihood of outbreaks in a city or town is related to the size of the human population; (viii) introduction pathways by road from Western Australia (for Medfly) and eastern Australia (for QFF) are shown to exist and to illegally or accidentally carry considerable amounts of fruit into South Australia; and (ix) there was no association between the numbers of either Queensland fruit fly or Medfly and the spatial pattern of either loquat or cumquat trees as sources of larval food in spring. This analysis supports the hypothesis that most fruit fly outbreaks in South Australia have been the result of separate introductions of infested fruit by vehicular traffic and that most of the resultant fly outbreaks were detected and died out within a few weeks of the application of eradication procedures. An alternative hypothesis, that populations of fruit flies are established in South Australia at below detectable levels, is impossible to disprove with conventional technology, but the likelihood of it being true is minimised by our analysis. Both hypotheses could be tested soon with newly developed genetic techniques.

Approximating persistence in a general class of population processes

We provide a general framework for estimating persistence in populations which may be affected by catastrophic events, and which are either unbounded or have very large ceilings. We model the population using a birth-death process modified to allow for downward jumps of arbitrary size. For such processes, it is typically necessary to truncate the process in order to make the evaluation of expected extinction times (and higher-order moments) computationally feasible. Hence, we give particular attention to the selection of a cut-off point at which to truncate the process, and we present a simple method for obtaining quantitative indicators of the suitability of a chosen cut-off. (c) 2005 Elsevier Inc. All rights reserved.